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A G-connection on E is an Ehresmann connection such that the parallel transport map τ : F x → F x′ is given by a G-transformation of the fibers (over sufficiently nearby points x and x′ in M joined by a curve). [5] Given a principal connection on P, one obtains a G-connection on the associated fiber bundle E = P × G F via pullback.
This definition may alternatively be expressed in terms of an arbitrary local frame of E but without considering coordinates on M. Then a vector-valued differential 2-form is expressed by differential 2-forms s 1, ..., s r and the connection is expressed by the connection 1-forms, a skew-symmetric r × r matrix of differential 1-forms θ α β.
For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve. An affine connection is typically given in the form of a covariant derivative , which gives a means for taking directional derivatives of vector fields ...
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The connection form is not tensorial because under a change of basis, the connection form transforms in a manner that involves the exterior derivative of the transition functions, in much the same way as the Christoffel symbols for the Levi-Civita connection. The main tensorial invariant of a connection form is its curvature form.
If the principal bundle P is the frame bundle, or (more generally) if it has a solder form, then the connection is an example of an affine connection, and the curvature is not the only invariant, since the additional structure of the solder form θ, which is an equivariant R n-valued 1-form on P, should be taken into account.
There is another simple way of constructing the connection form ω using the embedding of M in E 3. [61] The tangent vectors e 1 and e 2 of a frame on M define smooth functions from E with values in R 3, so each gives a 3-vector of functions and in particular de 1 is a 3-vector of 1-forms on E. The connection form is given by
Local connectedness is, by definition, a local property of topological spaces, i.e., a topological property P such that a space X possesses property P if and only if each point x in X admits a neighborhood base of sets that have property P. Accordingly, all the "metaproperties" held by a local property hold for local connectedness. In particular: